高收敛率的GRQI型迭代法

Some GRQI-Type Schemes with High Convergence Rate

对广义特征值问题Ａｘ＝λＢｘ，Ａ，Ｂ∈Ｃｎ×ｎ（１），本文提出ｌ级ＨＧＲＱＩ格式，其中ｌ为任一自然数，它的局部收敛阶为ｌ＋１。当ｌ＝１时，它就是文［１］中所述的ＧＲＱＩ格式，如果用Ｇａｕｓｓ消元法解有关线性方程组，则当１＜ｌ＜＜ｎ时，ｌ级ＨＧＲＱＩ在每个迭代步中的运算量与ＧＲＱＩ的运算量基本持平。本文又将适用于普通特征值问题Ａｘ＝λｘ的Ｏｓｔｒｏｗｓｋｉ双边送代法（ＯＴＩ）推广到ｌ级ＨＧＯＴＩ，它适用于问题（１），且具有局部收敛率ｌ十１。当ｌ＝１且（１）中的Ｂ＝Ｉ时，ＨＧＯＴＩ便成了ＯＴＩ。ＨＧＯＴＩ与ＨＧＲＱＩ有类似的优点。

In the paper, for generalized eigenvalue problem Ax=λBx, A,B∈C￣n×n (1). the l-thgrad HGRQI and HGOTI schemes are proposed respectively, where l is a given natural number. Under some conditions, they both have local convergence rate l+1. The well-known GRQI[1] is just the l-st grad HGRQI. If the Gauss elimination method is used to solve relevant linearsystems, then when 1 < l<<n, in each iteration step, the flop's order required by the l-th gradHGRQI is about the same as one required by the GRQI. The case is similar to above betweenthe l-st grad HGOTI and the l-th grad HGOTI.